an:05552748 Zbl 1171.65014 Towers, John D. Finite difference methods for approximating Heaviside functions EN J. Comput. Phys. 228, No. 9, 3478-3489 (2009). 00248659 2008
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65D32 41A55 41A25 41A63 Heaviside function; level set method; quadrature; irregular region; singular source term; finite difference; regular grid; convergence rate; one-step algorithm; two-step algorithm; numerical examples The paper presents two algorithms to evaluate the integral $$\mathcal{I}:=\int_\Omega f(\vec{x})d\vec{x}$$, where $$\vec{x}\in \mathbb{R}^n$$, $$\Omega=\{\vec{x}:\,u(\vec{x})>0\}$$ and $$\partial\Omega=\{\vec{x}:\,u(\vec{x})=0\}$$ is a compact manifold of codimension one. The method to be employed is motivated by the expression $$\mathcal{I}=\int_{\mathbb{R}^n} H(u(\vec{x}))f(\vec{x})d\vec{x}$$, where $$H$$ is the Heaviside function. The approach consists of approximating $$H$$ by finite differencing its first few primitives, a technique already used by the author to approximate delta functions. A brief presentation of the algorithms is given below. Let $I(z)=\int_0^z H(\zeta)d\zeta\quad\text{and}\quad J(z)=\int_0^z I(\zeta)d\zeta.$ The following relationships are derived: $I(u)=\langle\nabla J(u),\nabla u \rangle /|\nabla u|^2,\quad H(u)=\langle\nabla I(u),\nabla u \rangle /|\nabla u|^2,$ where $$\langle \cdot,\cdot \rangle$$ stands for the inner product. By discretizing $$H(u)$$ the one-step algorithm $$FDMH_1$$ is obtained which converges at a rate of $$\mathcal{O}(h^2)$$ when $$u$$ is smooth enough. By discretizing both relationships the two-step algorithm $$FDMH_2$$ is derived which can converge at a rate of $$\mathcal{O}(h^3)$$. These results are validated by means of some numerical examples. Jesus Ill??n Gonz??lez (Vigo)